Murach A. Elliptic operators and Hormander spaces

A theory of linear elliptic operators and elliptic boundary-value problems in Hilbert scales of the Hormander spaces of functions/distributions of positive and negative smoothness is built in the thesis. The scales of the Hormander spaces on a closed smooth manifold are introduced, and their interpolation properties are studied. The new a priori estimates for solutions to the linear elliptic systems are proved, and the Fredholm property of the matrix elliptic operators is established on these scales. The local smoothness of the solutions is studied in the Hormander spaces and in the Holder spaces of integer order. In terms of the Hormander spaces, we get some new sufficient conditions for convergence of spectral expansions in eigenfunctions of the normal elliptic operators almost everywhere and in the Holder spaces of integer order. The new theorems on a solvability of the elliptic boundary-value problem on the scales of the Hormander spaces are proved. We find a condition on the space of right-hand sides of the elliptic equations under which the elliptic boundary-value problem operator is bounded and Fredholm on the corresponding couples of distribution spaces.